| L(s) = 1 | + (−1 − i)3-s + (−1 − i)7-s − i·9-s + 4·11-s + (3 − 3i)13-s + (3 − 3i)17-s + 6i·19-s + 2i·21-s + (−3 + 3i)23-s + (−4 + 4i)27-s + 2·29-s − 6i·31-s + (−4 − 4i)33-s + (3 + 3i)37-s − 6·39-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.577i)3-s + (−0.377 − 0.377i)7-s − 0.333i·9-s + 1.20·11-s + (0.832 − 0.832i)13-s + (0.727 − 0.727i)17-s + 1.37i·19-s + 0.436i·21-s + (−0.625 + 0.625i)23-s + (−0.769 + 0.769i)27-s + 0.371·29-s − 1.07i·31-s + (−0.696 − 0.696i)33-s + (0.493 + 0.493i)37-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.342276971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.342276971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (3 - 3i)T - 23iT^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (9 + 9i)T + 47iT^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + (-9 + 9i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-3 - 3i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-7 + 7i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.450063801991801703020153875300, −8.149386727088358790209677125127, −7.62448348572277372206878590224, −6.45207936515420238596367940894, −6.21990880438484251376851501867, −5.27624821061915138889392810623, −3.86454480606741618614138680070, −3.37360755439257778888374860730, −1.61465793991709838519340038760, −0.64783116632890172521948894174,
1.30952014260074673879855641354, 2.70036591644518861487848003218, 3.97986816408030996002268447173, 4.50389945906390817817446593542, 5.65825239570499149942260611104, 6.28092662299837410457452354841, 7.02989514194210897978227581508, 8.223414305163055493358569399233, 8.981171024730539065000692184466, 9.585877442308488987326388676005
